This choice gives us two alternative forms of geometry in which the interior angles of a triangle add up to exactly 180 degrees or less, respectively, and are known as Euclidean and hyperbolic geometries.

Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids ( saddle surfaces ), hyperboloids (" wastebaskets "), hyperbolic geometry ( Lobachevsky's celebrated non-Euclidean geometry ), hyperbolic functions ( sinh, cosh, tanh, etc.

Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities.

Around 1830 though, the Hungarian János Bolyai and the Russian Nikolai Ivanovich Lobachevsky separately published treatises on a type of geometry that does not include the parallel postulate, called hyperbolic geometry.

His later work, starting around the mid-1970s, revealed that hyperbolic geometry played a far more important role in the general theory of 3-manifolds than was previously realised.

However, in spherical geometry and hyperbolic geometry ( where the sum of the angles of a triangle varies with size ) AAA is sufficient for congruence on a given curvature of surface.

In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries.

Thus he may be referring to some concrete thing, or incident, in his immediate environment by some symbolic-sounding, hyperbolic reference to transcendental events on the global scene.

It sets out Nimzowitsch's most important ideas, while his second most influential work, Chess Praxis, elaborates upon these ideas, adds a few new ones, and has immense value as a stimulating collection of Nimzowitsch's own games accompanied by his idiosyncratic, hyperbolic commentary which is often as entertaining as instructive.

Crochet has been used by mathematician Daina Taimina in order to create a version of the hyperbolic plane.

Position was calculated using a hyperbolic navigation system ( multilateration ) by comparing the phase difference of the radio signals received from several fixed stations.

Each station transmitted a continuous wave signal that, by comparing the phase difference of the signals from the Master and one of the Slaves, resulted in a set of hyperbolic lines of position called a pattern.

The bifurcations of a hyperbolic fixed point x < sub > 0 </ sub > of a system family F < sub > μ </ sub > can be characterized by the eigenvalues of the first derivative of the system DF < sub > μ </ sub >( x < sub > 0 </ sub >) computed at the bifurcation point.

A given constant time difference between the signals from the two stations can be represented by a hyperbolic line of position ( LOP ).

By determining the intersection of the two hyperbolic curves identified by this method, a geographic fix can be determined.

The hyperbolic character of biodiversity growth in the Phanerozoic can be similarly accounted for by a feedback between the diversity and community structure complexity.

Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the figure eight knot complement.

* Bi-directional delay line, a numerical analysis technique used in computer simulation for solving ordinary differential equations by converting them to hyperbolic equations

The interval, with length two, demarcated by the positive and negative units, occurs frequently, such as in the range of the trigonometric functions sine and cosine and the hyperbolic function tanh.

The reflecting pond at Wilson Hall also showcases a hyperbolic obelisk, designed by Dr. Wilson.

Note that, by convention, sinh < sup > 2 </ sup > x means ( sinh x )< sup > 2 </ sup >, not sinh ( sinh x ), and similarly for the other hyperbolic functions when used with positive exponents.

While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter k. Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean ; this is a task for the physical sciences.

In 1871, Felix Klein, by adapting a metric discussed by Arthur Cayley in 1852, was able to bring metric properties into a projective setting and was therefore able to unify the treatments of hyperbolic, euclidean and elliptic geometry under the umbrella of projective geometry.

The model for hyperbolic geometry was answered by Eugenio Beltrami, in 1868, who first showed that a surface called the pseudosphere has the appropriate curvature to model a portion of hyperbolic space, and in a second paper in the same year, defined the Klein model which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent, so that hyperbolic geometry was logically consistent if and only if Euclidean geometry was.

This is in contrast to solutions of the wave equation, and more general hyperbolic partial differential equations, which typically have no more derivatives than the data.

In contrast, sinh < sup >- 1 </ sup > x refers to the inverse function arsinh x and not to a reciprocal ( and again likewise for the other hyperbolic functions ).

An anti de Sitter space, in contrast, is a general relativity-like spacetime, where in the absence of matter or energy, the curvature of spacetime is naturally hyperbolic.

In contrast to Bolyai's work, Lobachevsky's work contained only hyperbolic geometry.

Wave equations are examples of hyperbolic partial differential equations ¸ but there are many variations.

Prior to Thurston, there were only a handful of known examples of hyperbolic 3-manifolds of finite volume, such as the Seifert – Weber space.

However, jazz historian, musician, and composer Gunther Schuller writes about Morton's " hyperbolic assertions " that there is " no proof to the contrary " and that Morton's " considerable accomplishments in themselves provide reasonable substantiation ".

However, in plane mapping there are two other angles to consider: the hyperbolic angle and the slope, which is the analogue of angle for dual numbers.

And, there exist C < sup > 1 </ sup > isometric embeddings of the hyperbolic plane in R < sup > 3 </ sup >.

In the first case, replacing the parallel postulate ( or its equivalent ) with the statement " In a plane, given a point P and a line ℓ not passing through P, there exist two lines through P which do not meet ℓ " and keeping all the other axioms, yields hyperbolic geometry.

In the hyperbolic model, within a two-dimensional plane, for any given line ℓ and a point A, which is not on ℓ, there are infinitely many lines through A that do not intersect ℓ.

Let there be these three hyperbolic paraboloids:

The notion was used as part of his hyperbolic doubt, wherein one decides to doubt everything there is to doubt.

There are no squares as such in the hyperbolic plane, although there are regular quadrilaterals, meaning quadrilaterals with all sides congruent and all angles congruent ( but these angles are strictly smaller than right angles ).

Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one line parallel to L passing through p. In elliptic geometry, there are no parallel lines at all.

When there is only one real root ( and p ≠ 0 ), it may be similarly represented using hyperbolic functions, as

If there exist one or more eigenvalues, then if the corresponding b < sub > i </ sub > ≠ 0, it is a paraboloid ( either elliptic or hyperbolic ); if the corresponding, the dimension i degenerates and does not get into play, and the geometric meaning will be determined by other eigenvalues and other components of b. When it is a paraboloid, whether it is elliptic or hyperbolic is determined by whether all other non-zero eigenvalues are of the same sign: if they are, then it is elliptic ; otherwise, it is hyperbolic.

In this instance the involution is termed " hyperbolic ", while if there are no fixed points it is " elliptic ".

It is then a theorem that there are in fact infinitely many such lines through P. Note that this axiom still does not uniquely characterize the hyperbolic plane uniquely up to isometry ; there is an extra constant, the curvature K < 0, which must be specified.

At the next roundabout with the A638 ( for Retford ) and B1164 ( for West Markham ) at Markham Moor ( again being grade-separated ) there is the Markham Hotel and two Little Chefs, including one ( originally a petrol station ) designed by Sam Scorer with a hyperbolic paraboloid-shaped roof that was destined to be demolished, although has survived and was listed by the Department for Culture Media and Sport in 2012.

In a hyperbolic space there is no limit to the number of spheres that can surround another sphere ( for example, Ford circles can be thought of as an arrangement of identical hyperbolic circles in which each circle is surrounded by an infinite number of other circles ).